Assume $M$ is an invertible positive matrix of rank $N$. The Von Neumann entropy $H$ of a matrix $M$ with eigenvalues $\{ \lambda_n \}$ is

$H[M] = -\sum_{n=1}^N \lambda_n \ln \lambda_n$.

In principle, the eigenvalues are encoded in the characteristic polynomial

$\phi_t (M) = \mathrm{det}(tI-M) = \prod_n(t-\lambda_n) = t^{N} + a_{N-1} t^{N-1} \cdots + a_1 t +a_0 $.

The trace $\mathrm{Tr}\, M$ is given by the coefficient $a_{N-1}$ in the characteristic polynomial:

$\lim\limits_{t \to \infty} \dfrac{\phi_t (M)-t^N}{t^{N-1}} = a_{N-1} = (-1)^{N-1} \, \mathrm{Tr}\, M = (-1)^{N-1} \sum_n \lambda_n$.

Is there a similar relationship between the entropy and the characteristic polynomial?